Complexity as a Discourse on School Mathematics Reform

• In Event: 34.078 - Roundtable Session 13
  In Roundtable Session: 34.078-3 - The Transdisciplinary Nature of Mathematics Education

Abstract

The intention of this paper is to present complexity science as, simultaneously, a curricular, an epistemological, and a pedagogical discourse. That is, (1) it can (and should) serve as the source of curriculum content, especially with regard to school mathematics, (2) it might be productively construed as a theory of learning when applied to the level of individual sense-making, and (3) it offers advice for teaching when applied to the phenomena of collective activity and knowledge production.

This paper is both informed by complexity science, which might be (incompletely) defined as the study of phenomena that arise in, but transcend, the co-specifying activities of their elements or subsystems. The domain encompasses a broad range of emphases and methods, and it is argued in this paper that educational researchers must be aware of and responsive to these diversities of interpretation – embracing both “hard complexity” (i.e., well bounded phenomena that lend themselves to quantification and digital modeling) and “soft complexity” (i.e., typically not well bounded, able to be mimicked but not to be confidently modeled).

The discussion of complexity is embedded in a report of a teaching experiment that was developed around the topic of exponentiation. Using principles of design research (and consistent with the methodology’s rooted in complexity thinking), this work involved iterative cycles of design, relied on the collaboration of diversely positioned agents, utilizes multiple data-gathering methods (Barab & Squire, 2004). For the elements of the project reported in this paper, data sources include video records of in-class events, artifacts generated collectively and individually by learners, and researcher-and-teacher recollections.

The report is organized around three “roles” that complexity science might play in school mathematics. Firstly, complexity is viewed in curricular terms for how it might affect the content of school mathematics. In this case, exponentiation (interpreted as an iterative operation) served as a focal point for understanding the nature of change in complex systems – in particular, for making sense of how the same operation can generate both minor and explosive change. Secondly, complexity is presented as a discourse on learning, which might influence how topics and experiences are formatted for students. In this regard, the report will highlight a portion of the teaching experiment in which exponentiation was contrasted with addition and multiplication, foregrounding the role and power of analogy in the development of mathematical understanding. Thirdly, complexity is interpreted as a source of pragmatic advice for teaching mathematics by interpreting design element of the lesson in terms of conditions necessary to complex emergence, as outlined by complexity researchers.

Complexity research is establishing a larger and larger footprint in the field of education. Fuller appreciations of its diverse interpretations, its span of application, and its entailments for schooling are vital for engaging with and extending the discourse.

Author

• Brent Davis, University of Calgary